Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/467

On Weber's Formula, considered as resulting from an Action transmitted from one Electric Particle to the other with a Constant Velocity.

KEYSTONE OF ELECTRODYNAMICS

861.] In a very interesting letter of Gauss to W. Weber^[1] he refers to the electrodynamic speculations with which he had been occupied long before, and which he would have published if he could then have established that which he considered the real keystone of electrodynamics, namely, the deduction of the force acting between electric particles in motion from the consideration of an action between them, not instantaneous, but propagated in time, in a similar manner to that of light. He had not succeeded in making this deduction when he gave up his electrodynamic researches, and he had a subjective conviction that it would be necessary in the first place to form a consistent representation of the manner in which the propagation takes place.

Three eminent mathematicians have endeavoured to supply this keystone of electrodynamics.

862.] In a memoir presented to the Royal Society of Gottingen in 1858, but afterwards withdrawn, and only published in Poggendorff's Annalen in 1867, after the death of the author, Bernhard Riemann deduces the phenomena of the induction of electric currents from a modified form of Poisson's equation

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \rho ={\frac {1}{a^{2}}}{\frac {d^{2}V}{dt^{2}}},$

where $V$ is the electrostatic potential, and $a$ a velocity.

This equation is of the same form as those which express the propagation of waves and other disturbances in elastic media. The author, however, seems to avoid making explicit mention of any medium through which the propagation takes place.

The mathematical investigation given by Riemann has been examined by Clausius^[2], who does not admit the soundness of the mathematical processes, and shews that the hypothesis that potential is propagated like light does not lead either to the formula of Weber, or to the known laws of electrodynamics.

863.] Clausius has also examined a far more elaborate investigation by C. Neumann on the 'Principles of Electrodynamics' ^[3]. Neumann, however, has pointed out^[4] that his theory of the transmission of potential from one electric particle to another is quite different from that proposed by Gauss, adopted by Riemann, and criticized

1. March 19, 1845, Werke, bd. v. 629.
2. Pogg., bd. cxxxv. 612.
3. Tubingen, 1868.
4. Mathematische Annalen, i. 317.